The great importance of the concepts of rigid body and rigid connection in Newtonian mechanics, is to the closest related with the fundamental views concerning space and time. Because the requirement that lengths shall be mutually comparable at different times, directly leads to the formation of the concept of measuring rods whose length is independent of time and motion, i.e., which are rigid. Later, this concept of the rigid body proves to be fruitful for the development of dynamics itself; because the rigid body as a continuous mass system of only six degrees of freedom, is not only kinematically of the highest simplicity, but also dynamically by allowing the composition [ 2 ] of the forces – which are acting at its points – to "resulting" forces and moments of the same magnitude, whose knowledge is sufficient for the description of motion. All these possibilities are principally based on the Galilei-Newtonian connection of space and time into a four-dimensional manifold[WS 1] (which I will call "world" following Minkowski[1]); a connection essentially contained in the theorem, that the natural laws not only shall be independent from the choice of the origin and the unit of time, as well as from the location of the spatial reference system and the unit of length, but also from a uniform translation given to the reference system under maintenance of the measure of time.

Exactly these foundations of kinematics are the ones to be abandoned, when the electrodynamic relativity principle – as stated by Lorentz, Einstein, Minkowski and others – comes into play. Because here, the connection of space and time into the "world" is different: the independence of the natural laws from the uniform translation of the spatial reference system only then takes place, when also the time parameter experiences a change, which not only tantamounts to a displacement of the origin and the choice of another unit. It is most closely connected to this, that measuring rods that maintain their length at uniform translation in the co-moving coordinate system, suffer a contraction in the direction of their velocity when viewed from a stationary system. By that, the concept of the rigid body fails, at least in its form adapted to Newtonian kinematics.

However, a corresponding concept is by no means to be dispensed with in the new kinematics as well, since otherwise the comparison of lengths of moving bodies at different times becomes illusory. No difficulty arises at the formation of this concept for systems moving relative to each other, [ 3 ] and the authors (mentioned above) of the foundational works on this theory, are using this circumstance without giving a particular definition of rigidity.

The difficulty only then arises, when accelerations are present. Only one attempt exists – made by Einstein[2] –, without completely clarifying the subject. Therefore I have undertaken the elaboration of the kinematics of the rigid body on the basis of the relativity postulate. Its possibility is probable from the outset, because the Newtonian kinematics represents in every relation a limiting case of the new kinematics, namely that one in which the speed of light c{\displaystyle c} is seen as infinitely great. The method used by me, consists in defining rigidity by a differential law instead of an integral law.

Indeed, one arrives in this way at the general rigidity conditions in differential form, which are very analogous to the corresponding conditions of the old kinematics and also pass into them for c=∞{\displaystyle c=\infty }. The integration of these conditions, which is very easily executable in the old kinematics in general, and which leads to the constancy of the distance of rigidly connected points, was only executed by me for the case of uniformly accelerated translation; the result is hardly inferior to the old kinematics in terms of simplicity and illustratability, and makes the assumption near at hand, as to what may be the result at arbitrary curvilinear and rotatory motions; though I'm not discussing this. The main result is (at uniform motion), that the motion of a single point of a rigid body co-determines the motion of all other ones by a very simple law, i.e., that the body thus only has one degree of freedom.

Now the question arises, whether (as in the old mechanics) the rigid body has simple properties in its dynamic behavior also in the new kinematics, and of course it will be about electromagnetic forces.

[ 4 ]The practical value of the new definition of rigidity must therefore prove itself in the dynamics of the electron; the greater or lesser clarity of the results achieved there, is to be used to a certain degree also in favor or against the acceptance of the relativity principle per se, since experiments have probably given no definite decision and maybe won't give one.

The theory of Abraham, which studies the motion of an electron (being rigid in the ordinary sense) in the force field produced by itself, has not only led to a qualitatively satisfying explanation of the phenomena of inertia of free electrons on a pure electric basis, but has also led to a quantitative law for the dependency of the electrodynamic mass from velocity at very small accelerations, which is probably not to be seen as disproved by the experiments. Though this theory which superimposes the rigid body (which is suited to the old mechanics) upon electrodynamics, doesn't satisfy the relativity principle, and this is the reason why its further development – at which Sommerfeld,[3]P. Hertz,[4]Herglotz,[5]Schwarzschild[6] and others are participating – leads to extraordinary mathematical complications. Now, already Lorentz tried to adapt Abraham's theory to the relativity principle, and for that purpose he constructed his "deformable" electron. Exactly this electron is to be denoted as rigid according to the definition given by me. That despite of this agreement, Lorentz's theory gives rise to contradictions to which Abraham[7] has alluded, is due to the fact that the laws of the composition of forces at the rigid body into resulting forces, were taken over without criticism from the old mechanics; [ 5 ] as to how these laws are to be modified, will be given by itself in the representation chosen here. Lorentz's formula for the dependency of mass from velocity, which represent the experiments as good as Abraham's formula, proves to be correct also in the more strict theory. Because this law, as it was already noticed by Einstein, and which was discussed by me in the paper[8] concerning "the inertial mass and the relativity principle" for arbitrary currents, is a direct consequence of kinematics and is not at all essentially connected with the actual electrodynamic mass, the "rest mass".

Yet, my theory strictly provides the dependency of the rest mass on acceleration for a class of motions, which corresponds – being the principally simplest accelerated motions – to the uniformly accelerated ones of the old mechanics, and which I call "hyperbolic motions", namely the rest mass proves to be constant up to enormous accelerations. Equations of motion in the form of the mechanical fundamental equations[9] which are adapted to the relativity principle, apply to this motion. Yet, since every accelerated motion can be approximated by such hyperbolic motions as long as their acceleration doesn't vary too suddenly, one achieves in this way an electrodynamic foundation of the fundamental equations of mechanics. This theory fails only for very rapidly changing accelerations; then also radiation resistances arise besides the inertial resistances. It is remarkable, that an electron causes no actual radiation, as great as its acceleration may be, but it drags its field along with it, which was up to now only known for uniformly moving electrons. The radiation and the resistance of the radiation only arise at deviations from hyperbolic motion.

My rigidity definition proves to be appropriate for the system of Maxwell's electrodynamics quite in the same [ 6 ] way, as the old definition of rigidity for the system of Galilei-Newtonian mechanics. The rigid electron in this sense, represents the dynamically most simple motion of electricity. One can even go so far to assert, that the theory provides clear hints to an atomistic structure of electricity, which is not at all the case in Abraham's theory. Thus my theory is in agreement with the atomistic instinct of so many experimentalists, for which the interesting attempt of Levi-Civita[10] – to describe the motion of electricity as a freely moving fluid being bound by no kinematic conditions – will hardly find applause.

Since the simplicity of the dynamics is therefore not inferior to the simplicity of kinematics of the new rigid body, then one will ascribe to this concept of rigidity the same fundamental importance in the system of the electromagnetic world-picture, as the ordinary rigid body in the system of the mechanical world-picture.

With respect to the electrodynamic applications of the second and the third chapter, we won't concern ourselves with rigid systems of discrete points, but with continuous rigid bodies. A continuous current of matter can be represented in the way named after Lagrange, by giving the space coordinates x,y,z{\displaystyle x,y,z} as functions of time t{\displaystyle t} and of three parameters ξ,η,ζ{\displaystyle \xi ,\eta ,\zeta } – for example the values of x,y,z{\displaystyle x,y,z} at time t=0{\displaystyle t=0}:

is orthogonal[11]; i.e., when A¯{\displaystyle {\overline {A}}} denotes the transposed matrix of A{\displaystyle A}, and 1 is the unit matrix, then it is

(4)

A¯A=1{\displaystyle {\overline {A}}A=1}

Fig. 1.

To oversee the generalization capability of this condition of the kinematics of the relativity principle, it is advantageous to use the interpretation (used by Minkowski in the work just cited) of the variables x,y,z,t{\displaystyle x,y,z,t} as parallel coordinates in a space of four dimensions called "world". In the following, the figures shall always mean the plane cut y=0,z=0{\displaystyle y=0,\ z=0} through a four-dimensional space; within them, we draw the x{\displaystyle x}-axis [ 8 ] horizontally, and the t{\displaystyle t}-axis upwards. The path of a point is represented in the xyzt{\displaystyle xyzt}-manifold (world) as a curve, the "world line", and the motion of a body is represented by a bundle of world lines. The previous condition dr/dt=0{\displaystyle dr/dt=0} now means, that the connecting line of the passage points (of two world-lines each) through a three-dimensional structure t{\displaystyle t} = const., has the same length for all those structures. Thus it is related to the three-dimensional points t{\displaystyle t} = const. "parallel" to space t=0{\displaystyle t=0}.

The importance of this rigidity condition for Newtonian mechanics, lies in the fact that it is invariant with respect to transformations, which transfer the Newtonian equations of motion into themselves. These transformations have the form, when the origin is maintained:

where kαβ,kα{\displaystyle k_{\alpha \beta },\ k_{\alpha }} are constants, and the matrix

K=(kαβ){\displaystyle K=\left(k_{\alpha \beta }\right)}

is orthogonal:

(6)

K¯K=1{\displaystyle {\overline {K}}K=1}

The orthogonal constituent only denotes the passage from the initial coordinate system to a system rotated around the origin; yet the second part denotes a uniform translation in time. This is represented in our four-dimensional world as passage from the initial t{\displaystyle t}-axis to an inclined t¯{\displaystyle {\overline {t}}}-axis. One immediately sees (Fig. 1), that the quantity r{\displaystyle r} indeed remains unchanged at this occasion.

The relativity principle of electrodynamics states an invariance of natural laws with respect to other linear substitutions, and by that the meaning of quantity r{\displaystyle r} becomes irrelevant. These "Lorentz transformations" connect the four magnitudes x,y,z,t{\displaystyle x,y,z,t} with four new ones x¯,y¯,z¯,t¯{\displaystyle {\overline {x}},{\overline {y}},{\overline {z}},{\overline {t}}} by such linear equations

At first it seems impossible as well, to provide an analogous condition between two world-lines, since there are no three-dimensional spaces with respect to transformations (7), (8), which are so preferred as previously the spaces t{\displaystyle t} = const. with respect to (5).

Therefore for the sake of generalization, one has to look after another definition of rigidity in the old kinematics. For that, one can use the circumstance that one can replace condition r{\displaystyle r} = const. (taking place between two finitely distant world lines) by a differential condition between infinitely adjacent world lines, so that, when the differential condition is satisfied in the whole space, it gives rise to equation r{\displaystyle r} = const.

[ 10 ] For that purpose, we consider at time t{\displaystyle t} the distance of two infinitely adjacent world lines, i.e., the line element

ds=dx2+dy2+dz2{\displaystyle ds={\sqrt {dx^{2}+dy^{2}+dz^{2}}}\,}

If one sets this equal to a constant ϵ{\displaystyle \epsilon }, then equation

ds2=ϵ2{\displaystyle ds^{2}=\epsilon ^{2}\,}

represents an infinitely small sphere. This emerges from an infinitely small ellipsoid during the motion represented by (1), which one obtains when one represents the quantity ds2{\displaystyle ds^{2}} by means of the equations

Now, we will call this motion only in the smallest parts as rigid, when an infinitely small structure is not changed during motion, thus when all pαβ{\displaystyle p_{\alpha \beta }} are independent of time. Thus we have the infinitesimal rigidity conditions:

In the following, only such quantities shall have a meaning, which are invariant with respect to the Lorentz transformations (7), (8).

Now we consider a current, which we represent instead of equations of form (1), by the following equations which better correspond to the symmetry of quantities x,y,z,t{\displaystyle x,y,z,t} required by the relativity principle:

where we confine ourselves to terms being linear in the increments dξ,dη,dζ{\displaystyle d\xi ,\ d\eta ,\ d\zeta } (which are first to be imagined as small, though finite). Here, x,y,z,t{\displaystyle {\mathfrak {x,y,z,t}}} are the functions defined by (16), and it is set:

Two spacetime vectors with components x1,y1,z1,t1{\displaystyle x_{1},y_{1},z_{1},t_{1}} and x2,y2,z2,t2{\displaystyle x_{2},y_{2},z_{2},t_{2}} are called normal, when their direction are conjugated with respect to the invariant hyperbolic structure

All vectors being normal to a time-like vector[14]x1,y1,z1,t1{\displaystyle x_{1},y_{1},z_{1},t_{1}} are satisfying a three-dimensional linear structure, which can be made to space t=0{\displaystyle t=0} by a suitable Lorentz transformation; we call it the normal cut of the vector.

[ 13 ] The concepts being so defined, are evidently invariant with respect to Lorentz transformations.

Now we consider a certain point P{\displaystyle P} upon the world line ξ=η=ζ=0{\displaystyle \xi =\eta =\zeta =0}, belonging to the value τ0{\displaystyle \tau _{0}} of the proper time. Through this point P{\displaystyle P}, we lay the normal cut to the velocity vector x0′,y0′,z0′,t0′{\displaystyle {\mathfrak {x'_{0},y'_{0},z'_{0},t'_{0}}}} in P{\displaystyle P}:

We can see this as an equation for τ{\displaystyle \tau }, from which one can calculate the values of proper time τ{\displaystyle \tau } (belonging to normal cut τ0{\displaystyle \tau _{0}}) upon the neighboring line dξ,dη,dζ{\displaystyle d\xi ,d\eta ,d\zeta }. Since the difference τ−τ0=dτ{\displaystyle \tau -\tau _{0}=d\tau } is small, then (21) will be a linear equation in dτ{\displaystyle d\tau }. Namely, if one expands

This cuts the normal cut (20) into a figure, which is to be seen as the "rest shape" of the filament at this place.

If we accordingly replace in (27), x,y,z,t{\displaystyle x,y,z,t} by expressions (17), and then the quantities x,y,z,t,xξ,…{\displaystyle {\mathfrak {x,y,z,t}},x_{\xi },\dots } by the expansions (22), it follows

By that, the rest shape is given as a quadratic form in dξ,dη,dζ{\displaystyle d\xi ,d\eta ,d\zeta }. Since point ξ=η=ζ=0{\displaystyle \xi =\eta =\zeta =0}, τ=τ0{\displaystyle \tau =\tau _{0}} was an arbitrary point of the current, one can omit indices 0 and replace x′…{\displaystyle {\mathfrak {x}}'\dots } by xτ,…{\displaystyle x_{\tau },\dots }. If we then write (29) in the form

then the rectangular matrix for 4 rows and 3 columns C=(cαβ){\displaystyle C=\left(c_{\alpha \beta }\right)} is equal to the product of two matrices S{\displaystyle S} and A{\displaystyle A}, which are formed from the derivatives of functions (14):

With the aid of equation (15) causing the determinant of S{\displaystyle S} to vanish, this relation can still further be simplified; namely it is easily given by computation:

(36)

S¯S=S{\displaystyle {\overline {S}}S=S}

and (33) thus goes over into:

(37)

P=A¯SA{\displaystyle P={\overline {A}}SA}

This is analogues to equation (12) derived in § 1. The six quantities pαβ{\displaystyle p_{\alpha \beta }} are to be denoted as "deformation quantities", and would be of importance in a theory of elasticity adapted to the relativity principle.

We will call a filament as rigid in the smallest parts, whose rest shape is independent from proper time τ{\displaystyle \tau }, i.e., for which the six equations

[ 16 ]When these equations are satisfied in the whole space, then we are dealing with the motion of a rigid body.

By that, we have gained the general differential conditions of rigidity. Since they are solely formed by the aid of such concepts being invariant with respect to Lorentz transformations, they thus have necessarily the same property.

This can be formulated in two ways. According to the one of Euler, one sees ϱ,wx,wy,wz{\displaystyle \varrho ,w_{x},w_{y},w_{z}} as functions of x,y,z,t{\displaystyle x,y,z,t}; then the continuity equation reads:

If we now denote (in the scheme of determinant D{\displaystyle D}) by S(∂xα/∂ξβ){\displaystyle S\left(\partial x_{\alpha }/\partial \xi _{\beta }\right)} the sub-determinant belonging to ∂xα/∂ξβ{\displaystyle \partial x_{\alpha }/\partial \xi _{\beta }}, then [ 18 ] it is given by successive differentiation of equations (14) with respect to xα{\displaystyle x_{\alpha }}, and by solving of the linear equations emerging in this way:

Formulas (46), (47), (50) have invariant character with respect to Lorentz transformations

The quantity

ϱ∗D=ϱ0{\displaystyle \varrho ^{*}D=\varrho _{0}}

is only dependent from ξ,η,ζ{\displaystyle \xi ,\eta ,\zeta }; when D{\displaystyle D} is equal to 1 for τ=0{\displaystyle \tau =0} (which one can always assume), then ϱ0{\displaystyle \varrho _{0}} is the "initial value of the rest density".

A current is called incompressible in the old kinematics, when ϱ{\displaystyle \varrho } is constant and independent from time t{\displaystyle t}. In the new kinematics we will define it as follows:

A current is incompressible, when the rest density ϱ∗{\displaystyle \varrho ^{*}} is constant, i.e., independent from proper time τ{\displaystyle \tau }.

Two forms of the incompressibility condition are given from (46) and (50).

We now want to integrate the differential conditions of rigidity (38) for the simplest case of uniform translation. When we consider, that rigidity must be identical with incompressibility in this case, then we not only achieve by that a criterion as to what our rigidity definition means mutatis mutandis, but simultaneously also a method for integration.

Thus we set

(55)

y=η,z=ζ{\displaystyle y=\eta ,\ z=\zeta }

and assume that x{\displaystyle x} and t{\displaystyle t} only depend on ξ{\displaystyle \xi } and τ{\displaystyle \tau }. Then we obtain from (32) and (33):

The simplest solution of these equation is obtained, when one puts φt{\displaystyle \varphi _{t}} and φx{\displaystyle \varphi _{x}} equal to constants γ{\displaystyle \gamma } and −δ{\displaystyle -\delta }, which must satisfy the condition

where W{\displaystyle W} and V{\displaystyle V} are two arbitrary functions of ξ{\displaystyle \xi }. Due to equation (63), the form of equations (64) is indeed conserved, when x,t{\displaystyle x,t} are subjected to a Lorentz transformation.

Equations (64) together with (55) represent a rectilinear uniform motion. Functions W(ξ),V(ξ){\displaystyle W(\xi ),\ V(\xi )} are determined by the value, which x{\displaystyle x} and t{\displaystyle t} shall have for τ=0{\displaystyle \tau =0}. It is not convenient here to assume x=ξ{\displaystyle x=\xi } for τ=0{\displaystyle \tau =0}, but functions W(ξ),V(ξ){\displaystyle W(\xi ),\ V(\xi )} are so to be determined, that formulas (64) represent that Lorentz transformation which transforms the body into rest, i.e., it is to be set:

As soon as one of the two quantities φt,φx{\displaystyle \varphi _{t},\ \varphi _{x}} depends on t{\displaystyle t} in (62), then this must also be the case for the other one. In this case, the integration of (62) can be easily executed by the aid of a Legendre transformation. [ 22 ] Then, one can namely introduce the quantity

(67)

φt=p{\displaystyle \varphi _{t}=p}

as independent variable besides x{\displaystyle x}, and imagine t{\displaystyle t} [from (67)] to be calculated as a function of x{\displaystyle x} and p{\displaystyle p}. If one then introduces (instead of φ{\displaystyle \varphi }) the new unknown function

where w{\displaystyle w} means an arbitrary function. From that it follows by differentiation with respect to q{\displaystyle q} under consideration of (68):

(70)

pc2qx−w′(p)=−t{\displaystyle {\frac {p}{c^{2}q}}x-w'(p)=-t}

If one consequently imagines p{\displaystyle p} as being calculated as a function of t{\displaystyle t} and inserted into φ=ψ+pt{\displaystyle \varphi =\psi +pt}, then one has the desired most general solution of (62):

from which it follows that any equation φ=const.=−ξ{\displaystyle \varphi =\mathrm {const} .=-\xi } represents the world line of a point of the rigid body. From (70) and (71) we consequently find the following representation of the world lines:

Here, the world lines of the rigid body are so described, that x{\displaystyle x} and t{\displaystyle t} are given as functions of the independent variables ξ,p{\displaystyle \xi ,\ p}. We now want to discuss this representation.

First it is to be noticed, that the rectilinear translatory motion only depends on one arbitrary function of an argument w(p){\displaystyle w(p)}. Thus one can say, that also here only one degree of freedom (as in the old kinematics) is present. There, the usage of the independent variables p=xτ{\displaystyle p=x_{\tau }} is essential, which still will be of great importance. Furthermore, equations (73) go over into the corresponding representation of the uniform translation of the old kinematics, when c=∞{\displaystyle c=\infty }. Because q=1+(p2/c2){\displaystyle q={\sqrt {1+\left(p^{2}/c^{2}\right)}}} becomes equal to 1 in this case; from the second equation (73) it follows for c=∞{\displaystyle c=\infty }, that p{\displaystyle p} only depends on t{\displaystyle t}, so that the first one assumes the form

x=ξ+a(t){\displaystyle x=\xi +a(t)}

Finally we direct our attention to the characterization of the world lines in the xt{\displaystyle xt}-plane. One recognizes, that (72) and (73) have the form of a Lorentz transformation and their inverse ones, which transform the variables x,y{\displaystyle x,y} into the variables x¯=w−ξ{\displaystyle {\overline {x}}=w-\xi }, t¯=qw′{\displaystyle {\overline {t}}=qw'}, and they read:

because the equations (66) between the coefficient are evidently satisfied due to (60).

Thus we are faced with a bundle of Lorentz transformations depending on the parameter p{\displaystyle p}. The motion, or rather the corresponding bundle of world lines, can now be described as follows.

If one gives a certain value ξ1{\displaystyle \xi _{1}} to ξ{\displaystyle \xi }, then x{\displaystyle x} and t{\displaystyle t} are given by equations (73) as specific functions of p{\displaystyle p}, which represent the world line of point ξ1{\displaystyle \xi _{1}}. The [ 24 ] components of the velocity world-vector with respect to axes x{\displaystyle x} and t{\displaystyle t}, are p{\displaystyle p} and −q{\displaystyle -q}. By that single curve ξ1{\displaystyle \xi _{1}}, all curves of the bundle are co-determined. One has to construct it as follows: To the tangent in a point p{\displaystyle p} of the curve, one lays the line being normal to it in the sense of § 2 (p. 12); this forms (together with the tangent) the x{\displaystyle x}- and t{\displaystyle t}-axis of a transformed coordinate system. Upon this x{\displaystyle x}-axis, one draws the distance ξ1−ξ{\displaystyle \xi _{1}-\xi } in the unit of this coordinate system[16]. If one now moves this coordinate system along curve ξ1{\displaystyle \xi _{1}}, then point ξ{\displaystyle \xi } follows the world line belonging to the parameter value ξ{\displaystyle \xi }. All points of such a normal (x{\displaystyle x}-axis) belong to the same value p{\displaystyle p}, thus they have the same velocity.

Fig. 4.

The uniform motion of a rigid body is so constituted, that – as soon as one point is transformed to rest – all of its points are transformed to rest by the same transformation. This rest transformation is exactly (74). The lines of same velocity p{\displaystyle p} = const., except at uniform motion, always have an envelope; the regularity of motion stops at this one. At given dimensions of the body, the curvature of world lines thus cannot exceed a certain limit, and vice versa. From that it follows, that a rigid body is necessarily extended into all directions, and has to be the smaller, the bigger the accelerations are that it should experience. Here, we have the first hint at the fundamental importance of atomistics in the new dynamics. If the rigid body carries a substance of rest density ϱ∗{\displaystyle \varrho ^{*}}, then it is independent of p{\displaystyle p}, and it is a function of ξ,η,ζ{\displaystyle \xi ,\eta ,\zeta }, which we denote by

From that one recognizes, that the corresponding world line in the xt{\displaystyle xt}-plane and the planes y=η,z=ζ{\displaystyle y=\eta ,\ z=\zeta } being parallel to it, are hyperbolas having the lines corresponding to the speed of light as asymptotes, and which are cutting the x{\displaystyle x}-axis in the distance ξ{\displaystyle \xi } from the origin. A bundle of such hyperbolas represents a motion, at which the rigid body comes from infinity and is approaching the origin, then it is turning back and is moving away into infinity again, where its velocity decreases from the speed of light to zero first, and after the turning back increases again to c{\displaystyle c}. This motion, to some extent analogous to the uniformly accelerated motion of old kinematics, we want to call hyperbolic motion shortly.

represent no essentially different motion; only the velocity is then different from zero for t=0{\displaystyle t=0}. Thus we will be able to confine ourselves to formulas (75), (76).

[ 26 ] This hyperbolic motion proves to be not only kinematically, but also dynamically as the simplest one. This is closely connected with the circumstance, that any world line is osculated by a hyperbola in any of its points P{\displaystyle P}, the "curvature hyperbola", where the vector of magnitude b=c2/ξ{\displaystyle b=c^{2}/\xi } directed from its center to point P{\displaystyle P}, represents the acceleration vector of the world line in P{\displaystyle P}.

Indeed, if we calculate the acceleration components of hyperbolic motion, then we find at first

is the magnitude of acceleration. From (81), (81), (82), the previous assertion follows.[17]

The acceleration is thus constant for every world line of hyperbolic motion in terms of their magnitude; here lies the analogy with the uniformly accelerated motion of old mechanics represented by parabolic world lines. Thus it is the simplest accelerated motion, and every motion can approximated by hyperbolic motions. [ 27 ] Based on that, we want to find out more precisely the dynamics of hyperbolic motions, above all we try to determine the force exerted by an electrically charged rigid body upon itself. The result (as approximation) will then also give information about all motions, in which the magnitude of the acceleration vector is only slightly changed.

Second chapter. The field of the rigid electron in hyperbolic motion.[edit]

The forces exerted by moving electric charges, which enter into the equations of motion of these charges, are derived from certain auxiliary quantities, the retarded potentials and field strengths. We want to summarize the expression for these quantities, which are employed in the following.

Let an electric current be represented by equations of form (14); let the initial value of its rest density (see § 3, p. 18, (51)) be:

where x¯,y¯,z¯,t¯{\displaystyle {\overline {x}},{\overline {y}},{\overline {z}},{\overline {t}}} and x¯τ,y¯τ,z¯τ,t¯τ{\displaystyle {\overline {x}}_{\tau },{\overline {y}}_{\tau },{\overline {z}}_{\tau },{\overline {t}}_{\tau }} denote the functions (14) or their derivatives with respect to τ{\displaystyle \tau }, taken from the arguments ξ¯,η¯,ζ¯{\displaystyle {\overline {\xi }},{\overline {\eta }},{\overline {\zeta }}}, and the function of x,y,z,t,ξ¯,η¯,ζ¯{\displaystyle x,y,z,t,{\overline {\xi }},{\overline {\eta }},{\overline {\zeta }}} is to be inserted for τ{\displaystyle \tau } into the brackets, which is given by solving the equation

with respect to τ{\displaystyle \tau }; namely that solution of the equation which is definitely determined [ 28 ] is to be taken[18], for which t>t¯{\displaystyle t>{\overline {t}}}. As to how the expressions (83), which are surely not used yet for continuous currents in this form, are connected to the ordinary formulas for the retarded potentials, shall be shortly explained in the next paragraphs.

The electric field strength E{\displaystyle {\mathfrak {E}}} and the magnetic one M{\displaystyle {\mathfrak {M}}} are derived from the potentials according to the vector equations:

[ 29 ] extended over an area G{\displaystyle G} of the xyzt{\displaystyle xyzt}-manifold, becomes an extremum, where the current of electricity and the values of the potentials are given upon the boundary of G{\displaystyle G}.

The expressions (83) for the potentials can be seen as the superposition of the elementary potential, stemming from the individually moving points of the current. For the latter, one namely has the expressions according to Liénard and Wiechert[21]

is the component of its velocity wx,wy,wz{\displaystyle w_{x},w_{y},w_{z}} in the direction of r{\displaystyle r}, and t¯{\displaystyle {\overline {t}}} instead of t{\displaystyle t} is to be set in the brackets, which is given from the equation

(93)

t−t¯=rc{\displaystyle t-{\overline {t}}={\frac {r}{c}}}

If one now has a continuous current, then world line (90) is to be replaced by a bundle of world lines, [ 30 ] by bringing function (90) into the form (1) by inserting three parameters ξ,η,ζ{\displaystyle \xi ,\eta ,\zeta }, and by replacing e{\displaystyle e} by density ϱ(ξ,η,ζ){\displaystyle \varrho (\xi ,\eta ,\zeta )}. Then, functions φx,φy,φz,φ{\displaystyle \varphi _{x},\varphi _{y},\varphi _{z},\varphi } become also dependent on ξ,η,ζ{\displaystyle \xi ,\eta ,\zeta }, and one can integrate them over the entire space. There it is also to be noticed, that it is

[ 31 ] and x¯,y¯,z¯,t¯=t−r/c{\displaystyle {\overline {x}},{\overline {y}},{\overline {z}},\ {\overline {t}}=t-r/c} are to be introduced as arguments in the brackets. The integrations in (95) are to be extended over all charges, i.e., over temporally variable boundaries, since they are in motion. The passage from expressions (95) to expressions (83) now exactly consists in this, that one brings the integrals to invariable limits independent of time. This has to be happening in the following way.

In the equations of current (1), we replace t{\displaystyle t} by t¯=t−r/c{\displaystyle {\overline {t}}=t-r/c}, then we obtain equations of the form

which connect x¯,y¯,z¯{\displaystyle {\overline {x}},{\overline {y}},{\overline {z}}} with ξ,η,ζ{\displaystyle \xi ,\eta ,\zeta } and evidently exactly represent the transformation, which brings the integrals to invariable limits when applied to (96); because this transformation (97) represents x¯,y¯,z¯{\displaystyle {\overline {x}},{\overline {y}},{\overline {z}}} as function of their initial values for the relevant instant in the brackets.

we want to denote the derivative from x¯{\displaystyle {\overline {x}}} to ξ{\displaystyle \xi } at maintained t¯{\displaystyle {\overline {t}}} with (∂x¯/∂ξ)t¯{\displaystyle (\partial {\overline {x}}/\partial \xi )_{\overline {t}}}, and at maintained t{\displaystyle t} with (∂x¯/∂ξ)t{\displaystyle (\partial {\overline {x}}/\partial \xi )_{t}}. If we differentiate then equations (97) (one after the other) with respect to ξ,η,ζ{\displaystyle \xi ,\eta ,\zeta }, then we obtain three equation systems with identical coefficients; for example at the differentiation with respect to ξ{\displaystyle \xi }:

If we also replace wx{\displaystyle w_{x}} by xτ/tτ{\displaystyle x_{\tau }/t_{\tau }} etc. as on p. 30, then formulas (107) go directly over into expressions (83). Indeed, only the initial density ϱ0{\displaystyle \varrho _{0}} occurs in them.

Since yτ=zτ=0{\displaystyle y_{\tau }=z_{\tau }=0}, then also Φy=Φz=0{\displaystyle \Phi _{y}=\Phi _{z}=0}. Since we have in (108) the quantity p{\displaystyle p} instead of τ{\displaystyle \tau } as independent variable besides ξ,η,ζ{\displaystyle \xi ,\eta ,\zeta }, we will see equation (84) h=0{\displaystyle h=0} as equation for p{\displaystyle p} as well. Then it reads:

p¯{\displaystyle {\overline {p}}} is to be calculated from these equations; namely that value of p¯{\displaystyle {\overline {p}}} is to be chosen, for which t>t¯{\displaystyle t>{\overline {t}}}. If one inserts the value following from the first equation

[ 35 ] Here, that sign is to be chosen, which corresponds to the smaller value of t¯{\displaystyle {\overline {t}}}. Now, since t¯=q¯ξ¯/c2{\displaystyle {\overline {t}}={\overline {q}}{\overline {\xi }}/c^{2}}, then the following is given (presupposed, that the electron is moving on the right side of the origin x=0{\displaystyle x=0}, i.e., ξ¯>0{\displaystyle {\overline {\xi }}>0}):

for all reference points, at which x/s>0{\displaystyle x/s>0}, the positive sign is to be taken,

for all reference points, at which x/s<0{\displaystyle x/s<0}, the negative sign is to be taken.

The distribution of these reference points can be derived from the figure.

Fig. 6.

We want to presuppose mostly in the following, that x/s>0{\displaystyle x/s>0}; only such points can be interior points of the electron at hyperbolic motion. Thus the positive square root B{\displaystyle B} is to be taken for them. When we sometimes also consider points for which x/s<0{\displaystyle x/s<0}, then we have to replace +B{\displaystyle +B} by −B{\displaystyle -B} everywhere.

From relations (118) we see, that the electron is carrying its field; the rest potentials observed by an observer co-moving with the electron, only depend on the rest coordinates ξ,η,ζ{\displaystyle \xi ,\eta ,\zeta }.

We still want to provide the explicit expression for the scalar rest potential Φ{\displaystyle \Phi }:

If the reference point is located in the area x/s<0{\displaystyle x/s<0}, then the negative sign is to be chosen instead of the positive one. Both Φ¯x{\displaystyle {\overline {\Phi }}_{x}} and Φ¯{\displaystyle {\overline {\Phi }}} become infinite at ξ=0{\displaystyle \xi =0}; the whole hyperbolic motion is singular for this value; yet the field strengths remain, as we will see, finite everywhere and are defined in the whole xyzt{\displaystyle xyzt}-manifold.

From that it follows, that Ex{\displaystyle {\mathfrak {E}}_{x}} only depends (besides η,ζ{\displaystyle \eta ,\zeta }) on s{\displaystyle s}, i.e., on ξ{\displaystyle \xi }, yet not on p{\displaystyle p}. The z{\displaystyle z}-component of the electric field strength is thus constant along every world line of the electron.

If one computes Ex{\displaystyle {\mathfrak {E}}_{x}}, then it is given: